A Mathematical model for Impact of citalopram on the HPA system and combined DEX/CRH test in 30 unipolar depressed patients.

T.Geetha. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 6, (Part -4) June 2017, pp.52-57
www.ijera.com DOI: 10.9790/9622-0706045257 52 | P a g e
A Mathematical model for Impact of citalopram on the HPA
system and combined DEX/CRH test in 30 unipolar depressed
patients.
T.Geetha *
, K.Sathishkumar **
*
Asst.Prof. of Mathematics, Kunthavai Naacchiyar Govt. Arts College for women, (Autonomous), Thanjavur-
613 007.Tamilnadu, India.
**
Asst.Prof. of Mathematics,Anjalai Ammal-Mahalingam Engineering College, Kovilvenni – 614 403.
Thiruvarur Dist., Tamilnadu, India.
ABSTRACT
In this paper we introduce bivariate sinh-normal distribution, which has seven parameters. Due to presence of
seven parameters it is a very flexible distribution .Different properties of this new distribution has been
established. We consider a bivariate model which can be obtained by transforming the sinh-normal distribution
and it is a generalization of the bivariate Birnbaum-Saunders distribution. Several properties of the bivariate
Birnbaum-Saunders distribution can be obtained as special cases of the proposed generalized bivariate
Birnbaum-Saunders distribution. Twenty three patients responded (≥50% reduction in the score), and
17 of them also reached criteria of remission (HDRS≤ 7). Baseline (dexamethasone suppressed) and CRH
stimulated ACTH concentrations significantly decreased from day 0 to day 28. CRH stimulated cortisol
concentrations also fell, although not significantly, but baseline cortisol concentrations exhibited a significant
increase from day 0 to day 28.Finally we conclude that the medical curve and Mathematical curve for disease
control is higher than the probability density functions which are monotonic functions.
Key words: ACTH, cortisol, Sinh-normal distribution, bivariate Birnbaum-Saunders distribution.
Mathematical subject AMA classification: 62
I. INTRODUCTION
Antidepressants are fundamental in the
treatment of depression [1]. However, all
antidepressants share a nonresponse rate of 30-50%.
Recent analyses show that there is no convincing
evidence that nonresponders to an antidepressant do
benefit from a switch to another antidepressant, as
compared to simply continuing the hitherto
ineffective antidepressant. This could in part be
explained by the fact that almost all antidepressants
share comparable neurobiological effects.I.e.
Enhancement of the serotonin and norepinephrine
neurotransmission in the CNS [2]. Therefore one
focus of psychiatric research is to develop agents for
the treatment of depression with different modes of
action. In this paper, the hypothalamic Pituitary
adrenocortical (HPA) system is of great interest,
because alterations in depressed subjects and changes
to the system under different treatment approaches
have repeatedly been demonstrated.
Frequently, the effects of antidepressive
compounds on the HPA system have been
investigated with the goal of understanding what
alterations in the HPA system regulation accompany
an improvement in depressive psychopathology. A
central hypothesis states that depression is typically
associated with a HPA system overdrive, which is cut
back in parallel to antidepressive treatment [4].
However, the results were partially inconsistent and
hence it appears likely that anti depressive substances
as well as subgroups of depression have to be
differentiated. Pariante for instance concluded in his
recent review that “drugs with opposite effects on the
HPA axis have antidepressant effects”, and that it is
neither conclusive “that high levels of
glucocorticoidsalways have a depressiogenic effect,
nor that decreasing the effects of these hormones
always has an anti depressant effect” [14]. This
resulted in the demand for personalized medicine
with treatments tailored to the specific pathology of
the patients [5]. As a result of this complexity,
researchers demanded more studies of the effects of
antidepressants on the HPA system
The combined dexamethasone/CRH test
(DEX/CRH test) is probably the most sensitive
challenge test of the HPA system in psychiatric
patients [6]. Only a few studies on the effects of
antidepressants with that particular method have been
conducted up to now [12], e.g.with tricyclic
antidepressants and with mirtazapine In most
countries SSRIs represent the most frequently
described class of antidepressants. Only two studies
up to now have investigated SSRI effects with
repeated applications of the DEX/CRH test. [13]
RESEARCH ARTICLE OPEN ACCESS

found a reduction of the ACTH and cortisol response
during treatment with citalopram as compared to the
pre SSRI situation. The other SSRI studied so far is
paroxetine.
We here present the results of a study of 30
unipolar depressed patients, examined with the
combined DEX/CRH test right before the start, and
after four weeks, of treatment with citalopram. We
chose citalopram, because besides its Senantiomer
escitalopram, citalopram is the most selective SSRI,
acting almost exclusively on the serotonin
transporter. This offers the opportunity to focus on
serotonergic effects on the HPA system regulation. In
addition, we aimed to include mainly drug naive
patients (i.e. without previous pharmacotherapy for
the index episode) to exclude inﬂuences of preceding
pharmacological treatment on the HPA system.
II. MATHEMATICAL MODELS:
The main aim of this paper is to introduce
bivariate sinh-normal (BSHN) distribution,which is a
natural extension of SHN distribution of one
dimension to two dimensions. Recently, Kundu et
al. [9] introduced a bivariate Birnbaum-Saunders
distribution. It is observed that the bivariate log-
Birnbaum-Saunders distribution can be obtained as a
special case of the BSHN distribution. The proposed
BSHN distribution has seven parameters, and due to
presence of two locations, two shapes, two scales and
one correlation parameters, it is a very flexible
model. After proper normalization, as the shape
parameters converge to zero, it approaches to a
standard bivariate normal distribution. The
probability density function (PDF) of BSHN,can be
both unimodal or bimodal depending on the shape
parameters. Due to presence of the two shape
parameters, it is more flexible than the bivariate
normal distribution. The marginals of BSHN are
SHN distributions and the conditional distribution
also can be obtained in closed form. We introduce a
new bivariate distribution, which can be obtained by
transforming the BSHN random variable. The new
distribution is a generalization of the bivariate
Birnbaum-Saunders distribution introduced by
Kundu et al.[9] and we call it as the bivariate
generalized Birnbaum-Saunders distribution. We
establish different properties of the generalized
Birnbaum-Saunders distribution and it is observed
that several properties of the bivariate Birnbaum-
Saunders distribution can be obtained as special cases
of the proposed distribution.
Suppose Y SHN (  ,, ) then the probability density function [10](PDF) becomes
       RyyAyayfY  ;,,;,,;,,; 
Here Ф (.) is the PDF of a standard normal random variable.
By simple transformation, it follows that if Y SHN (  ,, ), then
 1,0sinh
2
N
y
Z 




 




Here N (0, 1) denotes a standard normal random variable.
Rieck [15] has established the following properties of
a SHN distribution. The PDF of SHN is symmetric
about the location parameter µ.The distribution is
strongly unimodal for 2 and it is bimodal if
2 .It can be easily seen using L’Hospital’s rule,
that if
Y SHN (  ,, ) then
    01,0
2





asN
Y
It is an absolute continuous distribution with the PDF[11]
  0,2
2
1
exp
22
1
,; 2
2
3
2
1


































 t
t
t
tt
tfT





Here α is the shape parameter and β is the scale
parameter.
A standard BSHN random variable has the
PDF which is centred at (0,0) and it is symmetric
around (0,0).It is observed that the surface of the PDF
can be unimodal,bimodal or multimodal depending
on the parameter values.
III. APPLICATIONS:
3.1 ACTH
Mean baseline ACTH (before CRH
injection) declined from 14.9 pg/ml before citalopram
to 9.4 pg/ml after 4 weeks of treatment (p < 0.001,
Wilcoxon’s matched pairs test). With regard to the
ACTH values after CRH stimulation, a signiﬁcant
decrease during treatment was observed for the total
AUC as well (from 2077 to 1393, p = 0.001).

However, net AUC did not change significantly (p =
0.73), indicating that the main change during
treatment refers to an increased suppression effect to
dexamethasone, whereas the response to the CRH
stimulus was comparable between day 0 and day 28.
3.2 CORTISOL
Regarding cortisol, a significant increase in
the mean baseline values (before CRH injection) was
observed (from 14.5 ng/ml to 21.9 ng/ml, p = 0.03,
Wilcoxon’s matched pairs test), indicating a
reduction of the suppressive effect of dexamethasone
on the cortisol concentration during treatment with
citalopram. The CRH stimulated values (total AUC,
net AUC) remained unchanged.
3.3 CORTISOL/ACTH RATIO
The cortisol/ACTH ratio is a marker of
adrenal cortex responsiveness, indicating the amount
of cortisol that is produced in response to a defined
quantity of ACTH. According to the fact that the
ACTH parameters decreased while the cortisol
parameters tended to increase, a significant increase
in that ratio was detected for the baseline ratio and
for the total ratio (p = 0.00 ,p= 0.02 respectively),
indicating an increased responsiveness of the adrenal
gland to ACTH. Again, the net ratio did not change
significantly (p = 0.15), indicating that the main
changes from day 0 to day 28 concern the
suppressive effect of dexamethasone.
Fig 3.1(a)ACTH and Cortisol response (mean concentrations ± SEM) in the combined DEX/CRH test on the
day before citalopram treatment (baseline) and after four weeks of treatment (follow-up) (N=30)
IV. DISCUSSION
Thirty well characterized, mostly drug naive
patients participated in our study. With regard to
ACTH, the serum concentrations in the combined
dexamethasone/CRH test decreased after four weeks
of treatment with the SSRI citalopram. This is in line
with what has predominantly been observed during
antidepressant treatment in general and with what has
been observed in the two studies with repeated
DEX/CRH tests in SSRI treated patients conducted
so far. However, it is noteworthy that this decrease
particularly seems to be attributed to an improved
suppressive effect of dexamethasone. This is
indicated by the fact that only the baseline values (3
p.m., i.e. before CRH stimulation) and the total AUC
values, but not the net AUC values decreased
significantly from day 0 to day 28.
Regarding cortisol, the stimulating effect of
CRH decreased from day 0 to day 28 of citalopram
treatment indicated by a fall of net AUC from 3449
to 2932 (p = 0.4). Again, this is consistent with
results from other antidepressants .However, the
baseline cortisol concentrations (3 p.m.) increased
significantly during treatment, denoting that the
suppressive effect of dexamethasone declined.These
findings indicate that a blunting effect of the
citalopram treatment in the DEX/CRH test was only
present on the pituitary level of the HPA system,
whereas on the level of the adrenal gland a partial
activation was found, besides the good clinical
efficacy of citalopram in our study. Typically,
treatment with antidepressants and improvement in
depressive symptoms tends to be associated with a
blunting of the HPA system response. However, the
effect of antidepressants on the HPA system seems to
be temporarily associated with the activating effects
of most antidepressants in the beginning of the
treatment .In addition, recent findings and reviews
have pointed out that an activation of that system by
other substances trialed for treatment of depression
atleast does not hinder the relief of depressive
symptoms. [12] in their recent review concluded that
effective antidepressive pharmaceuticals might
alternatively lead to an activation or a blunting of
cortisol regulation, or even both. Their metaanalysis
revealed that approximately 56% of depressed
patients did not exhibit a decrease in their cortisol
concentration during treatment .Our group found a
HPA system activating effect in lithium
augmentation and in lithium monotherapy, despite a
good antidepressive outcome.
The increase in baseline cortisol (prior to
CRH injection) from day 0 to day 28 contrasts with

the drop in baseline ACTH. Since cortisol is released
by the adrenal gland primarily due to stimulition by
ACTH, alterations of the adrenal cortex have to be
taken into account. As also indicated by the
significant rise of the cortisol/ ACTH ratio from day
0 to day 28, an increase of the sensitivity of the
adrenal cortex has to be considered.A direct effect of
SSRIs on the adrenal gland has not been described up
to now. However, alterations of that gland associated
with a depressive episode and its regression have
repeatedly been reported. Imaging studies have
shown that the volume of the adrenal gland is
enlarged in depression (with a normalization after
recovery), presumably induced by excessive ACTH
stimulation. This continued overstimulation is
paralleled by an increase in the sensitivity of the
adrenal cortex to ACTH, resulting in a rise in the
cortisol/ACTH ratio. Depressed subjects exhibit a
greater cortisol response to the same amount of
ACTH than normal control subjects [8]. After CRH
stimulation, depressed subjects, in comparison to
healthy controls, repeatedly only showed an altered
ACTH response, but a similar cortisol response [3]
also indicating a modified ACTH response by the
adrenal gland. A low cortisol/ACTH ratio has been
associated with a response to lithium augmentation.
These facts point out that the responsiveness of the
adrenal gland to ACTH is regulated and flexible,
which includes the possibility that antidepressants
directly or indirectly have an impact on it. However,
due to the study design, our results unfortunately
cannot further elucidate this aspect.
With only 7 nonresponders, a comparison
with the 23 responders has methodological
limitations. Responders and nonresponders to the
citalopram treatment did not significantly differ with
regard to their DEX/CRH test results, neither before,
nor after 28 days of treatment. This indicates that the
DEX/CRH test was not helpful in predicting response
to the treatment. That means that, besides the fact that
this endocrinological test is rather too complex for an
everyday routine, the test does not provide adequate
support for a decision for or against citalopram
treatment. This is in line with other studies on the
predictive value of the DEX/ CRH test . However,
previous studies suggesting a predictive effect of the
DEX/CRH test reported a time lag between the
restoration of the HPA axis function and
psychopathological improvement. For instance, [7]
showed that a reduced cortisol response to the
DEX/CRH test is not associated with the current
depression state but with psycho pathological
improvement two to three weeks later. A number of
studies performing the DEX/CHR test in remitted
depression showed a predictive effect for relapse
during a follow up period of several months.
In addition, the two studies conducted so far
to prospectively investigate SSRI effects on the
DEX/CRH test included subjects who had failed to
respond to at least one antidepressant medication trail
for the current episode .It might well be that
treatment resistance and a more pronounced
depression severity on the one hand, and pretreatment
with antidepressants on the other hand, lead to
changes in the cortisol regulation differing to some
extent from those seen in the subjects of the study
presented here. The most pronounced alterations in
the HPA system were found in patients with severe
depression of the melancholic or psychotic subtype.
With 20 and 22 patients on SSRI respectively, these
two studies were somewhat smaller than the study
presented here (N = 30). Another reason for the small
cortisol effects in the current study might be related
with the sample selection, as it is well known that not
all patients with depression show an abnormal HPA
axis regulation, especially in an unselected sample of
depressed outpatients .In addition, with a mean of
43.8 (±13.8) years, the subjects reported here were
younger than the depressive patients in most of the
preceding studies applying the DEX/CRH test, which
were on average between 50 and 69 years old. [13]
studied younger patients (mean: 36 ± 9.6 years).
Higher age is associated with a more pronounced
alteration of the HPA system in depression. As with
nearly every study, this paper has methodological
strengths and limitations. On the positive side a small
drop out rate, a good characterization of the included
sample and an only minor proportion of participants
with pharmacological pretreatment for their
depressive index episode (17%) can be mentioned.
Established standardized clinical interviews and
rating scales were used to monitor and validate their
depressive symptoms. The neuroendocrinological
results were obtained by the best established and
validated test in this field, the combined DEX/CRH
test. With one exception, patients were not
hospitalized, thereby representing the majority of
patients treated with SSRIs. On the other hand, we
did not include a control group. A placebo
component would have been especially necessary to
reliably interpret the clinical efficacy of the
citalopram treatment and to evaluate if the
participants had indeed shown an overactive HPA
system before the start of treatment. However, this
was not the main objective of our study, which
primarily focused on the changes in cortisol
regulation induced by citalopram. Because the start
of citalopram was the only systematic change
between the first and the second DEX/CRH test, it is
rather unlikely that the differences found are due to
other factors than the SSRI and its therapeutic effects.
In addition, the time of the second DEX/CRH test
could be a matter of discussion. It cannot be excluded
that after a treatment period longer than four weeks a
decrease not only in ACTH, but also in the cortisol
parameters might have been demonstrated.

V. MATHEMATICAL RESULTS
VI. CONCLUSION
The blunting of the ACTH response in the
DEX/CRH test under citalopram is in line with what
has been observed in most studies with
antidepressants. However, the partial rise in cortisol
concentrations indicates an increase in the sensitivity
of the adrenal cortex to ACTH. State dependent
alterations in the volume and the ACTH
responsiveness of the adrenal gland have repeatedly
been reported in depressed subjects, which indicate
the possibility that SSRIs such as citalopram might
exhibit a direct or indirect effect on the adrenal
cortex. Twenty three patients responded (≥50%
reduction in the score), and 17 of them
also reached criteria of remission (HDRS≤ 7).
Baseline (dexamethasone suppressed) and CRH
stimulated ACTH concentrations significantly
decreased from day 0 to day 28. CRH stimulated
cortisol concentrations also fell, although not
significantly, but baseline cortisol concentrations
exhibited a significant increase from day 0 to day
28.Finally we conclude that the medical curve and
Mathematical curve for disease control is higher than
the probability density functions which are
monotonic functions.
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